#6467036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(33,40)(41,45)(42,46)(43,47)(44,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(19,27)(20,30)(21,33)(23,34)(24,37)(26,40)(28,31)(29,42)(32,43)(35,38)(36,46)(39,47);(9,12,13,11)(10,14,15,16)(17,19,21,27,22,29,33,42)(18,23,26,34,25,36,40,46)(20,31,30,44,32,28,43,41)(24,38,37,48,39,35,47,45);(17,20,21,30,22,32,33,43)(18,24,26,37,25,39,40,47)(19,28,27,41,29,31,42,44)(23,35,34,45,36,38,46,48);(1,3,4,7)(2,6,5,8)(18,25)(19,29)(24,39)(26,40)(27,42)(28,31)(37,47)(41,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374232);